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The Semantic Theory of Truth: Field’s Incompleteness Objection

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Abstract

According to Field’s influential incompleteness objection, Tarski’s semantic theory of truth is unsatisfactory since the definition that forms its basis is incomplete in two distinct senses: (1) it is physicalistically inadequate, and for this reason, (2) it is conceptually deficient. In this paper, I defend the semantic theory of truth against the incompleteness objection by conceding (1) but rejecting (2). After arguing that Davidson and McDowell’s reply to the incompleteness objection fails to pass muster, I argue that, within the constraints of a non-reductive physicalism and a holism concerning the concepts of truth, reference and meaning, conceding Field’s physicalistic inadequacy conclusion while rejecting his conceptual deficiency conclusion is a promising reply to the incompleteness objection.

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Notes

  1. Both reprinted in Tarski’s (1983) Logic, Semantics, Metamathematics. References to the former are from the reprint, to the latter are from the original.

  2. This means that it implies, for each declarative sentence of the language, a sentence of the form

    (T) s is true-in-L if and only if p,

    where ‘s’ has as substitution instances names of sentences in L and ‘p’ has as substitution instances translations of such object-language sentences. Insofar as it satisfies Convention T, then, the semantic definition of truth for the English language should yield as consequences the sentences

    (TI) ‘snow is white’ is true-in-English iff snow is white,

    (TI) ‘grass is green’ is true-in-English iff grass is green,

    and correspondingly for each declarative sentence of the language. For more on the material adequacy condition, otherwise referred to as ‘Convention T,’ see Tarski (1944), pp. 343–345.

  3. See Tarski (1944), pp. 343ff for more on how his definition satisfies such formal criteria.

  4. Later reprinted in Field (2001). All references are to the original.

  5. See, for example, the postscript to Field (1972) in Field’s (2001).

  6. According to Field, this is no minor accomplishment. One application of such a definition, Field contends, is its relevance to higher-order logic, i.e., that it can be used to clarify important concepts in model theory. See Field’s (1972), p. 351.

  7. Since Tarski defined satisfaction in terms of denotation and application, he did not require a freestanding definition of satisfaction.

  8. As Field notes, Tarski failed to reduce reference physicalistically in accordance with either the standards of so-called ‘strict reduction’ (whereby the bridging laws involve a high degree of precision) or ‘approximate reduction’ (whereby the bridging laws involve less precision). See Field’s (1972), pp. 357ff. For our purposes, the distinction can be ignored.

  9. See Field’s (1972), pp. 357–359, for a sensitive discussion of physicalism and its relation to IC, although Field’s formulation of this doctrine deviates slightly from my own. See also McDowell (1978), pp. 137ff concerning this issue.

  10. The valence of a chemical element is an integer associated with that element that represents the sorts of chemical combinations the element will enter into

  11. See Leeds (1978), pp. 120–123, for an illuminating discussion of this definition.

  12. In “An Alternative Reply” this rationale is called into question.

  13. Since these rejoinders were arrived at independently by Davidson and McDowell, they differ at least in point of detail. The focus here is on the common core of their arguments. See Stalnaker (1984) for rejoinders to IC similar to McDowell’s. See Field’s (2001, chapter 3) for a brief discussion of Stalnaker’s version of the incompleteness objection.

  14. An anti-realism about reference seems to follow more or less as a direct consequence: reference is not an existent relation between expressions and whatever expressions are supposed to provide deliverances about. See Davidson’s (1977), especially pp. 223–225, for more on the instrumentalist/anti-realist view of reference.

  15. I leave open the question of whether D&M were advancing a form of molecularism (according to which sentences or sub-sentential expressions are the primary bearers of meaning) or holism (according to which whole languages or theories are the primary bearers of meaning) as the basis of their reply to IC since this question does not impinge on the critique of D&M developed in the text.

  16. Such local holism should be kept distinct from the more radical thesis of global holism that is often alleged to be supported by Quinean arguments for the indeterminacy of reference/meaning: the thesis that the cognitive and explanatory significance of any expression E in a language L is holistically constrained by its connections with every other expression E in L.

  17. I am leaving the question open of whether our local holistic position will dictate that the semantic definition of truth is physicalistically incomplete. The possibility cannot be ruled out, though, that by giving word meaning an explanatory role, local holism might reinstate Field’s demand for a physicalistic explanation of reference; from our perspective, such an explanation would need to be non-reductive, of course.

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Acknowledgements

I would like to thank Gurpreet Rattan, Imogen Dickie, and anonymous referees for very helpful comments on this paper.

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Correspondence to Glen A. Hoffmann.

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Hoffmann, G.A. The Semantic Theory of Truth: Field’s Incompleteness Objection. Philosophia 35, 161–170 (2007). https://doi.org/10.1007/s11406-007-9057-9

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